Expected Value Questions on the ACT (and in real life)

In my last post I talked about a strategy of eliminating two answers that mean the same thing on the SAT writing/ ACT english section. Today, I plan to pivot abruptly and talk about a specific type of math concept seen often on the ACT: expected value.

Expected value questions often appear towards the end of the math section (questions 50-60) on the ACT and are usually accompanied by a table similar to the one below.

Score on Test

Probability

60

.10

70

.15

80

.30

90

.20

100

.25

The question usually asks something like “what is the expected score this student will get on this test.” These questions do use that specific word expected almost every time, which makes them easy to recognize!

First of all, here’s how most people try to solve this question. They add 60+70+80+90+100, and then divide by 5, as if they were finding a normal average. However, this doesn’t take into account that certain scores are more likely than other scores. For example, a student has a better chance of getting a 100 than getting a 60, but simply taking the averages of the scores doesn’t take this into account at all.

The method for solving expected value questions is unfamiliar to many but easy enough to learn. What you want to do is multiply each possible score by the probability that it happens, and then add up those products. Sounds harder than it is, so let me show you for this one.

60 x .10 = 6

70 x .15 = 10.5

80 x .30 = 24

90 x .20 = 18

100 x .25 = 25

And then we add those products: 6 + 10.5 + 24 + 18 + 25 = 83.5

And that’s it! 83.5 is the student’s expected score on this test!

Sometimes the chart will give fractions for the probabilities instead of decimals, but the process is exactly the same! Multiply each score by the fraction that represents its probability, and then add those up.

This is a great bit of math to know for the ACT, because these questions are usually featured in the “hard question” range, but are really among the easiest to set up and solve! On the SAT, a similar type of question appears in the form of histograms, but we’ll cover those another time.

Next time, I will show you examples of how calculating expected value actually has several applications to everyday situations, and hopefully convince a few more people that not all of this math stuff is totally irrelevant to their lives!